数学季刊 ›› 2020, Vol. 35 ›› Issue (4): 344-353.doi: 10.13371/j.cnki.chin.q.j.m.2020.04.002

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广义的Orlicz空间上的Korn不等式与散度方程

  

  1. 1. School of Longqiao College of Lanzhou University of Finance and Economics, Lanzhou 730101, China;  2. School of College of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing 404130, China
  • 收稿日期:2020-09-24 出版日期:2020-12-30 发布日期:2021-01-06
  • 作者简介:WU Rui-min(1985-), female, native of Lanzhou, Gansu, Longqiao College of Lanzhou University of Finance and Economics, engages in harmonic analysis; WANG Song-bai(1986-), male, native of Changsha, Hunan, professor of Chongqing Three Gorges University, Ph.D, engages in harmonic analysis.
  • 基金资助:
    Supported by the National Natural Science Foundation of China (Grant No.11726622);
    Scientific Research Fund of Young Teachers in Longqiao College (Grant No. LQKJ2020-01).

Korn’s Inequality and Divergence Equations on Generalize Orlicz Spaces

  1. 1. School of Longqiao College of Lanzhou University of Finance and Economics, Lanzhou 730101, China;  2. School of College of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing 404130, China
  • Received:2020-09-24 Online:2020-12-30 Published:2021-01-06
  • About author:WU Rui-min(1985-), female, native of Lanzhou, Gansu, Longqiao College of Lanzhou University of Finance and Economics, engages in harmonic analysis; WANG Song-bai(1986-), male, native of Changsha, Hunan, professor of Chongqing Three Gorges University, Ph.D, engages in harmonic analysis.
  • Supported by:
    Supported by the National Natural Science Foundation of China (Grant No.11726622);
    Scientific Research Fund of Young Teachers in Longqiao College (Grant No. LQKJ2020-01).

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摘要：

 Let ϕ be a generalized Orlicz function satisfying (A0), (A1), (A2), (aInc) and (aDec). We prove that the mapping

 f →f #:=supB 1/\int|B||f(x)-fB|dx is continuous on Lϕ(·)(Rn) by extrapolation. Based on this result we generalize Korn’s inequality to the setting of generalized Orlicz spaces, i.e., ||\triangledown f||L^{ϕ(·)}(Ω)  \lesssim||DF|||L^{ϕ}(Ω) . Using the Calder´on–Zygmund theory on generalized Orlicz spaces, we obtain that the divergence equation divu=f has a solution u∈(W01,ϕ(·)(Ω))n such that ||\triangledown u||L^{ϕ(·)}(Ω) \lesssim ||f||L^{ϕ}(Ω).

关键词:  Generalized Orlicz spaces, Korn’s inequality, Extrapolation.

Abstract:

 Let ϕ be a generalized Orlicz function satisfying (A0), (A1), (A2), (aInc) and (aDec). We prove that the mapping

 f →f #:=supB 1/\int|B||f(x)-fB|dx is continuous on Lϕ(·)(Rn) by extrapolation. Based on this result we generalize Korn’s inequality to the setting of generalized Orlicz spaces, i.e., ||\triangledown f||L^{ϕ(·)}(Ω)  \lesssim||DF|||L^{ϕ}(Ω) . Using the Calder´on–Zygmund theory on generalized Orlicz spaces, we obtain that the divergence equation divu=f has a solution u∈(W01,ϕ(·)(Ω))n such that ||\triangledown u||L^{ϕ(·)}(Ω) \lesssim ||f||L^{ϕ}(Ω).

Key words:  Generalized Orlicz spaces, Korn’s inequality, Extrapolation.

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